How many times do you forget a stranger?

Reddit user katyvs1 created an AskReddit thread with the following question: What’s a random statistic about yourself you’d love to know, but never will? I was excited to see that many of the responses were bona-fide Fermi questions, and one response in particular caught my imagination:

How many times have I walked past someone that I’ve walked past before without realising? — u/colecr

While we can’t give that user a relevant answer without knowing more about their life, it seems to me that we can come up with an estimate for the average American city dweller (from here on out referred to as Mike). The first step is to break the problem down into simpler questions.

1. How many people does Mike walk past every day?
2. How many people does Mike pass on more than one occasion?
3. How many people can Mike remember in total?

Photo by Ingo Joseph from Pexels

How many people does Mike walk past every day?

Let’s set a fairly realistic scene. Mike lives and works within a bustling neighborhood situated in a large city. We’ll call this neighborhood Fermitown. It’s roughly one square mile and holds about 20,000 residents.¹ Rather conveniently for this analysis, Fermitown is perfectly square, spanning twelve blocks in each dimension.² In total, Fermitown has 144 blocks and 576 sidewalks.

That’s more than enough sidewalk for Mike, who only walks about two miles on average.³ We’ll ignore street crossings and say that Mike walks 24 blocks each day, or 4.2% of the sidewalks. Like most city dwellers, Mike walks a relatively fast five feet per second. That means he spends 2112 seconds walking, or 2.4% of the day. Let’s keep it simple and say that for any given second of the day, he has a 0.1% chance of being on a specific sidewalk.

Remember that Mike is the everyman of his neighborhood. If we ignore the likelihood of popular paths or commute times we can use his statistic to estimate twenty residents on any sidewalk at any given time. From personal experience, that seems like a reasonable guess.⁴ So if we assume that half of those pedestrians are going the opposite direction of Mike, we can say that he passes ten residents per block.

Answer: Mike walks past 240 residents each day.

How many people does Mike pass more than once?

According to this model, Mike walks past 240 residents, or 1.2% of Fermitown’s population, every day. We can use that to approximate a 1.2% chance of seeing a specific resident on the sidewalk on any given day. After one year, Mike will have seen about 93% of the population more than once.⁵ After two years, assuming residents move every nine years, Mike will have seen about 21,900 faces at least twice, and a quarter of those faces at least ten times.

Answer: After two years Mike passes around 21,900 residents at least twice.

How many people can Mike remember?

Photo by jesse orrico on Unsplash

This is the hardest part of the question. The best statistic I can find is a recent study that pegs average recall at about 1,000 faces, and average recognition at about 5,000 faces (although at least one participant could recognize over 10,000 faces). The problem we face is that these numbers don’t tell us what Mike’s actual potential (i.e. facial recognition capacity) is. We can however assume that his capacity is partly used up by friends, family, significant figures, and celebrities. We also have to remember that we’re concerned with Mike’s ability to recognize faces that he passes on the street. In reality I suspect that multiple, significant encounters need to be had for a face to be committed to memory. We’re most likely overestimating the likelihood of Mike committing a face to memory and underestimating Mike’s ability to recognize faces in general, so I’ll leave the estimate at 5,000.

Answer: 5,000 faces.

Putting it all together

After two years of living in Fermitown, Mike has unknowingly passed by at least 16,900 people on more than one occasion.⁶ Assuming facial recognition capacity is limited to 5,000 faces, he would continue passing the rest of his neighborhood’s citizens multiple times without being the wiser. In reality I think Mike might perform much better than we’ve estimated. But even if he recognized 50% of the people he passed, that would still leave thousands unrecognized.

Let’s continue this example by moving Mike to the suburbs. Population density is much less, so we’ll say he lives in a town of 50,000 people and only walks past 15 strangers each day. Assuming he lives there for nine years, he’ll have seen nearly 10,500 residents at least twice. It took a little longer for multiple encounters to build up, but they eventually grew to twice our limit of 5,000 faces.

Final estimate: The average American will have unknowingly passed by thousands of strangers more than once. You can see my simulation’s code on GitHub. Let me know what I did wrong in the comments. (:

Footnotes

1. Using San Francisco as a guide, the average population per square mile is a little over 18,000. However, most neighborhoods have higher population densities.
2. I’m assuming a block length of 350 feet and street widths of 60 feet, using San Francisco as a guide once again.
3. Americans don’t seem to walk much at all.
4. It also implies that over half of the neighborhood’s residents are walking at any given time, which sounds less reasonable.
5. I originally tried a naive equation of 1-.988³⁶⁵ but it didn’t match my (most likely more accurate) simulation. In the meantime I’m going to look into a better way to model this distribution.
6. Remember this number only includes residents.